Several factors led to the development of mathematics in China being, for a long period, independent of developments in other civilisations. The geographical nature of the country meant that there were natural boundaries (mountains and seas) which isolated it. On the other hand, when the country was conquered by foreign invaders, they were assimilated into the Chinese culture rather than changing the culture to their own. As a consequence there was a continuous cultural development in China from around 1000 BC and it is fascinating to trace mathematical development within that culture. There are periods of rapid advance, periods when a certain level was maintained, and periods of decline.

The first thing to understand about ancient Chinese mathematics is the way in which it differs from Greek mathematics. Unlike Greek mathematics there is no axiomatic development of mathematics. The Chinese concept of mathematical proof is radically different from that of the Greeks, yet one must not in any sense think less of it because of this. Rather one must marvel at the Chinese approach to mathematics and the results to which it led.

Chinese mathematics was, like their language, very concise. It was very much problem based, motivated by problems of the calendar, trade, land measurement, architecture, government records and taxes. By the fourth century BC counting boards were used for calculating, which effectively meant that a decimal place valued number system was in use. It is worth noting that counting boards are uniquely Chinese, and do not appear to have been used by any other civilisation.

Our knowledge of Chinese mathematics before 100 BC is very sketchy although in 1984 the Suan shu shu (A Book on Arithmetic) dating from around 180 BC was discovered. It is a book written on bamboo strips and was found near Jiangling in Hubei province. The next important books of which we have records are a sixteen chapter work Suanshu (Computational prescriptions) written by Du Zhong and a twenty-six chapter work Xu Shang suanshu (Computational prescriptions of Xu Shang) written by Xu Shang. Neither of these texts has survived and little is known of their content. The oldest complete surviving text is the Zhoubi suanjing (Zhou Shadow Gauge Manual) which was compiled between 100 BC and 100 AD (see the article on The Ten Classics). It is an astronomy text, showing how to measure the positions of the heavenly bodies using shadow gauges which are also called gnomons, but it contains important sections on mathematics. It gives a clear statement on the nature of Chinese mathematics in this period (see for example [2]:-

The method of calculation is very simple to explain but has wide application. This is because a person gains knowledge by analogy, that is, after understanding a particular line of argument they can infer various kinds of similar reasoning ... Whoever can draw inferences about other cases from one instance can generalise ... really knows how to calculate... . To be able to deduce and then generalise.. is the mark of an intelligent person.

The Zhoubi suanjing contains a statement of the Gougu rule (the Chinese version of Pythagoras's theorem) and applies it to surveying, astronomy, and other topics. Although it is widely accepted that the work also contains a proof of Pythagoras's theorem, Cullen in [3] disputes this, claiming that the belief is based on a flawed translation given by Needham in [13].

In fact much Chinese mathematics from this period was produced because of the need to make calculations for constructing the calendar and predicting positions of the heavenly bodies. The Chinese word 'chouren' refers to both mathematicians and astronomers showing the close link between the two areas. One early 'choren' was Luoxia Hong (about 130 BC - about 70 BC) who produced a calendar which was based on a cycle of 19 years.

The most famous Chinese mathematics book of all time is the Jiuzhang suanshu or, as it is more commonly called, the Nine Chapters on the Mathematical Art. The book certainly contains contributions to mathematics which had been made over quite a long period, but there is little in the original text to distinguish the precise period of each. This important work, which came to dominate mathematical development and style for 1500 years, is discussed in the article Nine Chapters on the Mathematical Art. Many later developments came through commentaries on this text, one of the first being by Xu Yue (about 160 - about 227) although this one has been lost.

A significant mathematical advance was made by Liu Hui (about 220 - about 280) who wrote his commentary on the Jiuzhang suanshu or Nine Chapters on the Mathematical Art in about 263. Dong and Yao write [24]:-

Liu Hui, a great mathematician in the Wei Jin Dynasty, ushered in an era of mathematical theorisation in ancient China, and made great contributions to the domain of mathematics. From the "Jiu Zhang Suan Shu Zhu" and the "Hai Dao Suan Jing" it can be seen that Liu Hui made skilful use of thinking in images as well as in logical and dialectical ways. He solved many mathematical problems, pushing his mathematical reasoning further along the dialectical way.

Liu Hui gave a more mathematical approach than earlier Chinese texts, providing principles on which his calculations are based. He found approximations to π using regular polygons with 3 × 2ⁿ sides inscribed in a circle. His best approximation of π was 3.14159 which he achieved from a regular polygon of 3072 sides. It is clear that he understood iterative processes and the notion of a limit. Liu also wrote Haidao suanjing or Sea Island Mathematical Manual (see the article on The Ten Classics) which was originally an appendix to his commentary on Chapter 9 of the Nine Chapters on the Mathematical Art. In it Liu usesPythagoras's theorem to calculate heights of objects and distances to objects which cannot be measured directly. This was to become one of the themes of Chinese mathematics.

About fifty years after Liu's remarkable contributions, a major advance was made in astronomy when Yu Xi discovered the precession of the equinoxes. In mathematics it was some time before mathematics progressed beyond the depth achieved by Liu Hui. For example Sun Zi (about 400 - about 460) wrote his mathematical manual the Sunzi suanjing which on the whole provides little new. However, it does contains a problem solved using the Chinese remainder theorem, being the earliest known occurrence of this type of problem.

This text by Sun Zi was the first of a number of texts over the following two hundred years which made a number of important contributions. Xiahou Yang (about 400 - about 470) was the supposed author of the Xiahou Yang suanjing (Xiahou Yang's Mathematical Manual) which contains representations of numbers in the decimal notation using positive and negative powers of ten. Zhang Qiujian (about 430 - about 490) wrote his mathematical text Zhang Qiujian suanjing (Zhang Qiujian's Mathematical Manual) some time between 468 and 486. Its 92 problems illustrate the formula for summing an arithmetic progression. Perhaps it is most famous for presenting the 'Hundred fowls problem' which is an indeterminate problem with three non-trivial solutions.

One of the most significant advances was by Zu Chongzhi (429-501) and his son Zu Geng (about 450 - about 520). Zu Chongzhi was an astronomer who made accurate observations which he used to produce a new calendar, the Tam-ing Calendar (Calendar of Great Brightness), which was based on a cycle of 391 years. He wrote the Zhui shu (Method of Interpolation) in which he proved that 3.1415926 < π < 3.1415927. He recommended using ³⁵⁵/₁₁₃ as a good approximation and ²²/₇ in less accurate work. With his son Zu Geng he computed the formula for the volume of a sphere using Cavalieri's principle (see [25]). The beginnings of Chinese algebra is seen in the work of Wang Xiaotong (about 580 - about 640). He wrote the Jigu suanjing (Continuation of Ancient Mathematics), a text with only 20 problems which later became one of the Ten Classics. He solved cubic equations by extending an algorithm for finding cube roots. His work is seen as a first step towards the "tian yuan" or "coefficient array method" or "method of the celestial unknown" of Li Zhi for computing with polynomials.

Interpolation was an important tool in astronomy and Liu Zhuo (544-610) was an astronomer who introduced quadratic interpolation with a second order difference method. Certainly Chinese astronomy was not totally independent of developments taking place in the subject in India and similarly mathematics was influenced to some extent by Indian mathematical works, some of which were translated into Chinese. Historians argue today about the extent of the influence on the Chinese development of Indian, Arabic and Islamic mathematics. It is fair to say that their influence was less than it might have been, for the Chinese seemed to have little desire to embrace other approaches to mathematics. Early trigonometry was described in some of the Indian texts which were translated and there was also development of trigonometry in China. For example Yi Xing (683-727) produced a tangent table.

From the sixth century mathematics was taught as part of the course for the civil service examinations. Li Chunfeng (602 - 670) was appointed as the editor-in-chief for a collection of mathematical treatises to be used for such a course, many of which we have mentioned above. The collection is now called The Ten Classics, a name given to them in 1084.

The period from the tenth to the twelfth centuries is one where few advances were made and no mathematical texts from this period survive. However Jia Xian (about 1010 - about 1070) made good contributions which are only known through the texts of Yang Hui since his own writings are lost. He improved methods for finding square and cube roots, and extended the method to the numerical solution of polynomial equations computing powers of sums using binomial coefficients constructed with Pascal's triangle. Although Shen Kua (1031 - 1095) made relatively few contributions to mathematics, he did produce remarkable work in many areas and is regarded by many as the first scientist. He wrote the Meng ch'i pi t'an (Brush talks from Dream Brook) which contains many accurate scientific observations.

The next major mathematical advance was by Qin Jiushao (1202 - 1261) who wrote his famous mathematical treatise Shushu Jiuzhang (Mathematical Treatise in Nine Sections) which appeared in 1247. He was the first of the great thirteenth century Chinese mathematicians. This was a period of major progress during which mathematics reached new heights. The treatise contains remarkable work on the Chinese remainder theorem, gives an equation whose coefficients are variables and, among other results, Heron's formula for the area of a triangle. Equations up to degree ten are solved using the Ruffini-Horner method.

Li Zhi (also called Li Yeh) (1192-1279) was the next of the great thirteenth century Chinese mathematicians. His most famous work is the Ce yuan hai jing (Sea mirror of circle measurements) written in 1248. It contains the "tian yuan" or "coefficient array method" or "method of the celestial unknown" which was a method to work with polynomial equations. He also wrote Yi gu yan duan (New steps in computation) in 1259 which is a more elementary work containing geometric problems solved by algebra. The next major figure from this golden age of Chinese mathematics was Yang Hui (about 1238 - about 1298). He wrote the Xiangjie jiuzhang suanfa (Detailed analysis of the mathematical rules in the Nine Chapters and their reclassifications) in 1261, and his other works were collected into the Yang Hui suanfa (Yang Hui's methods of computation) which appeared in 1275. He described multiplication, division, root-extraction, quadratic and simultaneous equations, series, computations of areas of a rectangle, a trapezium, a circle, and other figures. He also gave a wonderful account of magic squares and magic circles.

Guo Shoujing (1231-1316), although not usually included among the major mathematicians of the thirteen century, nevertheless made important contributions. He produced the Shou shi li (Works and Days Calendar), worked on spherical trigonometry, and solved equations using the Ruffini-Horner numerical method. He also developed a cubic interpolation formula tabulating differences of the accumulated difference as in Newton's forward difference interpolation method.

The last of the mathematicians from this golden age was Zhu Shijie (about 1260 - about 1320) who wrote the Suanxue qimeng (Introduction to mathematical studies) published in 1299, and the Siyuan yujian (True reflections of the four unknowns) published in 1303. He used an extension of the "coefficient array method" or "method of the celestial unknown" to handle polynomials with up to four unknowns. He also gave many results on sums of series. This represents a high point in ancient Chinese mathematics.

The decline in Chinese mathematics from the fourteenth century was not by any means dramatic. The Nine Chapters on the Mathematical Art continued to be the model for mathematical learning and new works based in it continued to appear. For example Ding Ju published the Ding ju suan fa (Ding Ju's arithmetical methods) in 1355, He Pingzi published the Xiangming suan fa (Explanations of arithmetic) in 1373, Liu Shilong published the Jiu zhang tong ming suanfa (Methods of calculation in the 'Nine Chapters') in 1424, and Wu Jing published the Jiu zhang suan fa bi lei da quan (Complete description of the 'Nine Chapters') in 1450. Wu Jing was an administrator in the province of Zhejing and his arithmetical encyclopaedia contained all the 246 problems of the Nine Chapters. Again Cheng Dawei (1533 - 1606) published the Suanfa tong zong (General source of computational methods) in 1592 which is written in the style of the Nine Chapters on the Mathematical Art but provides an even larger collection of 595 problems.

The books we have just listed show mathematical activity, but they did not take forward the methods of polynomial algebra. On the contrary, the deep works of the 13^th century ceased to be even understood much less developed further. Xu Guangqi (1562 - 1633) certainly recognised exactly this and offered possible explanations including scholars neglecting practical computational tools and an identification of mathematics with mystical numerology under the Ming dynasty. Other factors must be that the books describing the advanced methods were, in the Chinese tradition, very terse, and without teachers to pass on an understanding it became increasingly difficult for scholars to learn directly from the texts. Xu Guangqi was the first native of China to publish translations of European books in Chinese. Collaborating with Matteo Ricci he translated Western books on mathematics, hydraulics, and geography. Certainly this does not mark the end of the Chinese mathematics tradition, but from the time of Matteo Ricci and other Western missionaries China was greatly influenced by other mathematical traditions.

It is impossible in an article of this length to mention many of the numerous contributions from this period on. Let us mention one important family, however, namely the Mei family. The most famous member of this family was Mei Wending (1633-1721) and his comment on the golden section is typical of the sensible attitude he took towards Western mathematics (see for example [9]):-

After having understood how to make use of the golden section, I began to believe that the different geometrical methods could be understood and that neither the missionaries attitude of considering this simple technique as a divine gift, nor the Chinese attitude of rejecting it as heresy is correct.

Mei chose not to take a government post as most mathematicians did, but rather decided to devote himself to mathematics and its teaching. He travelled widely throughout China and gained great fame leading to many people becoming his pupils. Two of his brothers, Mei Wenmi and Mei Wennai, worked on astronomy and mathematics. Mei Wending was assisted later in his life by his son Mei Yiyan. Mei Juecheng (1681-1763), who was Mei Wending's grandson, was asked in 1705 by Emperor Kangxi to be editor-in-chief of the major mathematical encyclopaedia Shuli jingyun (Collected basic principles of mathematics) (1723). Mei Juecheng also edited his grandfather Mei Wending's work producing the Meishi congshu jiyao (Collected works of the Mei family) in 1761.

Certain people from the eighteenth century onwards did an excellent job in recording the Chinese tradition so that much of it is still accessible to us today. For example Dai Zhen (1724 - 1777) became an editor for the Siku quanshu (Complete library of the four branches of literature) which was a project set up by Emperor Qianlong in 1773. He edited a new edition of the Nine Chapters on the Mathematical Art after copying the complete text as part of this project. Ruan Yuan (1764 - 1849) produced his famous work the Chouren zhuan or Biographies of astronomers and mathematicians containing biographies of 275 Chinese and 41 Western "mathematicians". Many biographical details of Chinese mathematicians recorded in this Archive are known through this work. Li Rui (1768 - 1817) assisted Ruan Yuan. He was a highly productive mathematician who died when at the height of his abilities. His most important work is Lishi suan xue yi shu (Collected mathematical works of Li Rui).

It is to the credit of Chinese mathematicians that they did not let their mathematical tradition be replaced by the western tradition. For example Li Shanlan (1811-1882) is important as a translator of Western science texts but he is most famous for his own mathematical contributions. He produced his own versions of logarithms, infinite series, and combinatorics which did not follow the style of western mathematics but his research naturally developed out of the foundations of Chinese mathematics. There were many other efforts to promote Chinese mathematics, and in particular a mathematics journal, the Suanxue bao, was set up in 1899. The editors wrote:-

Western methods should not be adulated and Chinese methods despised.

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